### Jackson Wrong and Hyperbolic Bessels

I'm back in the lab this week working on some numerical models for the solenoid to be used in the H-Ray experiment.  In looking for analytic solutions for the z and rho components of a magnetic field due to a coil, I came across a very interesting reference in an article by Bergeman, Erez, and Metcalf[1].  Take a look

If you're one of the many who after poring over Jackson, thought "This can't be right", in this one case, you were right.  The proper expressions according to Bergeman et al. are

Hyperbolic Bessels
Solving for quantized fields, one is often led to Bessel eigenfunctions in cylindrical coordinates.  Based on some of my recent work, I was left with the feeling that relativistic quantizations of the field in a uniformly accelerated space might give the result in terms of hyperbolic Bessel functions.  Sure enough, the Macdonald function solution that Dr. Fulling[2] finds in his paper "Nonuniqueness of Canonical Field Quantization in Riemannian Space-Time" is also known as the hyperbolic Bessel[3].  Cool!

References
1.Bergeman T., Erez G. & Metcalf H. (1987). Magnetostatic trapping fields for neutral atoms, Physical Review A, 35 (4) 1535-1546. DOI:

2. Fulling S. (1973). Nonuniqueness of Canonical Field Quantization in Riemannian Space-Time, Physical Review D, 7 (10) 2850-2862. DOI:

3.  http://en.wikipedia.org/wiki/Bessel_function#Modified_Bessel_functions_:_I.CE.B1.2C_K.CE.B1

### Cool Math Tricks: Deriving the Divergence, (Del or Nabla) into New (Cylindrical) Coordinate Systems

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The following is a pretty lengthy procedure, but converting the divergence, (nabla, del) operator between coordinate systems comes up pretty often. While there are tables for converting between common coordinate systems, there seem to be fewer explanations of the procedure for deriving the conversion, so here goes!

What do we actually want?

To convert the Cartesian nabla

to the nabla for another coordinate system, say… cylindrical coordinates.

What we’ll need:

1. The Cartesian Nabla:

2. A set of equations relating the Cartesian coordinates to cylindrical coordinates:

3. A set of equations relating the Cartesian basis vectors to the basis vectors of the new coordinate system:

How to do it:

Use the chain rule for differentiation to convert the derivatives with respect to the Cartesian variables to derivatives with respect to the cylindrical variables.

The chain rule can be used to convert a differe…

### Division: Distributing the Work

Our unschooling math comes in bits and pieces.  The oldest kid here, seven year-old No. 1 loves math problems, so math moves along pretty fast for her.  Here’s how she arrived at the distributive property recently.  Tldr; it came about only because she needed it.
“Give me a math problem!” No. 1 asked Mom-person.

“OK, what’s 18 divided by 2?  But, you’re going to have to do it as you walk.  You and Dad need to head out.”

And so, No. 1 and I found ourselves headed out on our mini-adventure with a new math problem to discuss.

One looked at the ceiling of the library lost in thought as we walked.  She glanced down at her fingers for a moment.  “Is it six?”

“I don’t know, let’s see,” I hedged.  “What’s two times six?  Is it eighteen?”

One looked at me hopefully heading back into her mental math.

I needed to visit the restroom before we left, so I hurried her calculation along.  “What’s two times five?”

I got a grin, and another look indicating she was thinking about that one.

I flashed eac…